Egg plates seem to have been developed exclusively for the purpose of serving Deviled Eggs, an hors d'oeuvre that came into popular use sometime after 1600 AD. Deviled eggs have their roots in ancient Roman recipes, and the first published recipes for stuffed, boiled eggs were from medieval Europe. By the 17th century this was a common way to prepare eggs, but it was not called "deviled" until the 18th Century, in England. The term "deviled" means a food seasoned with a spice mixture that usually includes mustard.

The egg plate may be an adaptation from oyster plates, which originated earlier but have larger hollows to hold the oysters. It is alleged that the Aetna Co. made an egg dish in 1884. I have not seen that dish, yet, so don't know if it really is an egg plate. 

The earliest made glass egg plates that I know about, so far, were made by von Streit in Germany. The earliest I know of in the United States was made by Duncan & Miller Glass Company. Their Early American Sandwich pattern was introduced around 1925, and the egg plate is shown in a 1943 catalog reprint. So we know that it was at least made in the 1940's. There is an egg plate done in their "First Love" etching, which has been found in the Pall Mall (circa 1940) and Canterbury (circa 1938) patterns and another egg plate is done in their "Passion Flower" Design with silver Intaglio etching (on a Pall Mall blank). 

Another Depression Era egg plate was made in the Candlewick pattern by Imperial, probably circa 1934. In 1948, the A. J. Heisey & Co. produced the Lariat pattern, which contains both an oval and a round egg plate. Most U.S. glass egg plates were made from the 1940's and later, but a few might have been made as early as the 1930's.